JNTU B.Tech II SIGNALS AND STOCHASTIC PROCESS Previous Question Paper (March - 2022 - R16)

Convolution is a fundamental mathematical operation that describes the output of a Linear Time-Invariant (LTI) system. In the time domain, the output y(t) of an LTI system with impulse response h(t) for an input x(t) is given by the convolution integral:

This process involves flipping the impulse response, shifting it by t, multiplying it by the input signal, and integrating over all time. It represents the accumulation of the system's responses to all infinitesimal impulses that compose the input signal.

In the frequency domain, convolution is simplified by the Convolution Theorem, which states that convolution in the time domain corresponds to multiplication in the frequency domain. If X(f) and H(f) are the Fourier transforms of x(t) and h(t) respectively, the output Y(f) is:

This transformation is computationally efficient, as it replaces complex integration with point-by-point multiplication of frequency spectra, which is the basis for fast convolution algorithms like FFT-based filtering.

For a system to provide distortionless transmission, the output signal y(t) must be an exact replica of the input signal x(t), possibly delayed and scaled in amplitude. Mathematically, this is expressed as:

where K is a constant gain and t₀ is a constant time delay. Applying the Fourier transform to both sides, we obtain:

Since the transfer function of the system is defined as H(f) = Y(f) / X(f), we get:

From this expression, we derive the two necessary conditions for distortionless transmission:

1. Amplitude condition: The magnitude response |H(f)| must be constant for all frequencies within the signal bandwidth, i.e., |H(f)| = K.
2. Phase condition: The phase response ∠H(f) must be a linear function of frequency, i.e., ∠H(f) = -2πft₀, which implies a constant group delay.